What is an Example of a Problem on a River Width?

S6-SA2-0134

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

A problem on river width usually involves using trigonometry or geometry to calculate the unknown width of a river. This is done by taking measurements from one bank and using angles and distances to find the width indirectly. It's like finding a hidden distance without directly crossing it.

Simple Example

Quick Example

Imagine you are standing on one side of a cricket ground and want to know the distance to the opposite boundary without walking across. If you know the angle to a specific point on the boundary and the distance to another point along your side, you can use these measurements to calculate the boundary's width. Similarly, for a river, you use angles and distances from one bank.

Worked Example

Step-by-Step

Let's say you want to find the width of a river. You stand at point A on one bank directly opposite a tree (point T) on the other bank. You then walk 50 meters along your bank to point B. From point B, you measure the angle to the tree (T) as 30 degrees. What is the width of the river (AT)?

1. Identify the knowns: Distance AB = 50 meters, Angle ABT = 90 degrees (since A is directly opposite T, and B is along the bank), Angle ATB (from B to T) = 30 degrees.
---2. Recognize the triangle: We have a right-angled triangle ABT, with the right angle at A.
---3. Choose the correct trigonometric ratio: We need to find AT (opposite side to angle B) and we know AB (adjacent side to angle B). So, tan(Angle B) = Opposite/Adjacent = AT/AB.
---4. Substitute the values: tan(30 degrees) = AT / 50.
---5. Calculate tan(30 degrees): We know tan(30 degrees) is approximately 1/sqrt(3) or 0.577.
---6. Solve for AT: AT = 50 * tan(30 degrees) = 50 * 0.577 = 28.85 meters.
---Answer: The width of the river is approximately 28.85 meters.

Why It Matters

Understanding how to calculate river width is crucial for civil engineers designing bridges or dams, and for environmental scientists studying water flow. It's also used in surveying and mapping, helping create accurate maps for various applications, even for planning smart cities or for defense purposes.

Common Mistakes

MISTAKE: Assuming the angle measured from the bank to the object on the other side is always 90 degrees. | CORRECTION: The 90-degree angle is usually formed between the line representing the river width and the line along the bank you walk on. The angle to the object is typically different.

MISTAKE: Using the wrong trigonometric ratio (e.g., using sine instead of tangent). | CORRECTION: Carefully identify which sides are 'opposite', 'adjacent', and 'hypotenuse' relative to the known angle in the right-angled triangle, then choose SOH CAH TOA accordingly.

MISTAKE: Not drawing a clear diagram of the situation. | CORRECTION: Always draw a simple diagram first, labelling all known distances, angles, and the unknown quantity. This helps visualize the right-angled triangle and avoids confusion.

Practice Questions

Try It Yourself

QUESTION: You are standing at point P on one bank of a river. Directly opposite you on the other bank is a big rock, R. You walk 40 meters along your bank to point Q. From Q, the angle to the rock R is 45 degrees. What is the width of the river (PR)? | ANSWER: 40 meters

QUESTION: A surveyor stands at point X on a river bank, directly opposite a tree Y. He walks 60 meters downstream to point Z. From Z, the angle to the tree Y is 60 degrees. Find the width of the river (XY). (Use sqrt(3) = 1.732) | ANSWER: Approximately 103.92 meters

QUESTION: To find the width of a river, a person stands at point A on one bank directly opposite a pole P on the other bank. They walk 30 meters along the bank to point B. From B, they sight the pole P and find the angle PBA to be 30 degrees. If they then walk another 20 meters to point C (so BC = 20m) and sight the pole again, what would be the new angle PCA? (Assume AB is perpendicular to AP). | ANSWER: Approximately 21.8 degrees

MCQ

Quick Quiz

Which trigonometric ratio is most commonly used to find the width of a river when you know the distance walked along the bank and the angle to the opposite object?

Sine

Cosine

Tangent

Secant

The Correct Answer Is:

Tangent (tan) is used because it relates the opposite side (river width) to the adjacent side (distance walked along the bank) in a right-angled triangle. Sine uses the hypotenuse and Cosine also uses the hypotenuse.

Real World Connection

In the Real World

Imagine the Border Roads Organisation (BRO) or an engineering team from L&T needing to build a bridge across a wide river in a remote area. They can't just stretch a measuring tape across. They use surveying instruments like a theodolite or total station from one bank to measure angles and distances to points on the opposite bank, then apply these trigonometric principles to accurately determine the river's width for bridge design.

Key Vocabulary

Key Terms

Trigonometry: A branch of mathematics dealing with the relations between the sides and angles of triangles. | Right-angled Triangle: A triangle with one angle exactly 90 degrees. | Angle of Elevation/Depression: The angle between the horizontal line of sight and the object when looking up or down. | Tangent (tan): A trigonometric ratio, in a right triangle, it's the ratio of the length of the side opposite an angle to the length of the side adjacent to the angle. | Surveying: The technique, profession, and science of determining the terrestrial or three-dimensional position of points and the distances and angles between them.

What's Next

What to Learn Next

Great job understanding river width problems! Next, you can explore 'Heights and Distances' problems, which are a direct extension of this concept. You'll learn to calculate the height of buildings or mountains using similar trigonometric methods, opening up more real-world applications!